Answer:
Mid-point: [tex](0,-5)[/tex]
Equation: [tex]y=\frac{7}{3}x-5[/tex]
Step-by-step explanation:
To find the mid-point of AB, simply add up their x and y coordinates and divide by 2 respectively to find their middle point.
[tex](\frac{{x__A}+{x__B}}{2},\frac{{y__A}+{y__B}}{2})[/tex]
[tex](\frac{{-7}+{7}}{2},\frac{{-2}+{(-8)}}{2})[/tex]
[tex](0,-5)[/tex]
To find the perpendicular slope that passes through the mid-point, we need to know the slope between AB first.
Slope of AB: [tex]\frac{{y__2}-{y__1}}{{x__2}-{x__1}}[/tex] = [tex]\frac{{-8}-{-2)}}{{7}-{(-7)}}[/tex] = [tex]\frac{-6}{14}[/tex] = [tex]\frac{-3}{7}[/tex]
Multiplying slopes that are perpendicular with each other always results in -1.
[tex]\frac{-3}{7}*m = -1[/tex]
[tex]m=\frac{7}{3}[/tex]
By the point slope form:
[tex]({y}-{y__1})=m({x}-{x__1})[/tex]
Plug in the coordinates of the mid-point:
[tex]({y}-{(-5)})=\frac{7}{3} ({x}-{0})[/tex]
Equation: [tex]y=\frac{7}{3}x-5[/tex]
